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Universidade Federal De Pernambuco Centro De Ciências Exatas E Da Universidade Federal de Pernambuco Centro de Ciˆencias Exatas e da Natureza Departamento de F´ısica P´os-gradua¸c˜ao em F´ısica VORTEX MOTION AROUND A CIRCULAR CYLINDER BOTH IN AN UNBOUNDED DOMAIN AND NEAR A PLANE BOUNDARY Marcel Nascimento de Moura DISSERTAC¸AO˜ DE MESTRADO Recife 17 de maio de 2012 Universidade Federal de Pernambuco Centro de Ciˆencias Exatas e da Natureza Departamento de F´ısica Marcel Nascimento de Moura VORTEX MOTION AROUND A CIRCULAR CYLINDER BOTH IN AN UNBOUNDED DOMAIN AND NEAR A PLANE BOUNDARY Trabalho apresentado ao Programa de P´os-gradua¸c˜aoem F´ısica do Departamento de F´ısica da Universidade Federal de Pernambuco como requisito parcial para obten¸c˜aodo grau de Mestre em F´ısica. Orientador: Giovani Lopes Vasconcelos Banca examinadora: Prof. Giovani Lopes Vasconcelos (Departamento de F´ısica - UFPE) Prof. Darren Gregory Crowdy (Department of Mathematics - Imperial College London) Prof. Cl´ecio Clemente de Souza Silva (Departamento de F´ısica - UFPE) Recife 17 de maio de 2012 Catalogação na fonte Bibliotecária Joana D’Arc L. Salvador, CRB 4-572 Moura, Marcel Nascimento de. Vortex motion around a circular cylinder both in an unbounded domain and near a plane boundary / Marcel Nascimento de Moura. – Recife: O Autor, 2012. xiv, 93 f.: fig. Orientador: Giovani Lopes Vasconcelos. Dissertação (Mestrado) - Universidade Federal de Pernambuco. CCEN. Física, 2012. Inclui bibliografia e apêndice. 1. Dinâmica dos fluidos. 2. Dinâmica de vórtices. 3. Sistemas hamiltonianos. I. Vasconcelos, Giovani (orientador). II. Título. 532.05 (22. ed.) FQ 2012-020 Universidade Federal de Pernambuco Departamento de Física – CCEN Programa de Pós-Graduação em Física Cidade Universitária - 50670-901 Recife PE Brasil Fone (++ 55 81) 2126-8449/2126-8450 - Fax (++ 55 81) 3271-0359 http://www.df.ufpe.br/pg e-mail: [email protected] Parecer da Banca Examinadora de Defesa de Dissertação de Mestrado Marcel Nascimento de Moura "VORTEX MOTION AROUND A CIRCULAR CYLINDER BOTH IN AN UNBOUNDED DOMAIN AND NEAR A PLANE BOUNDARY" A Banca Examinadora composta pelos Professores Giovani Lopes Vasconcelos (Presidente e Orientador), Clécio Clemente de Souza Silva, ambos do Departamento de Física da Universidade Federal de Pernambuco e Darren Greg Crowdy, do Imperial College, Londres, consideram o candidato: ( ) Aprovado ( ) Reprovado ( ) Em exigência Secretaria do Programa de Pós-Graduação em Física do Departamento de Física do Centro de Ciências Exatas e da Natureza da Universidade Federal de Pernambuco em dezessete de maio de dois mil e doze. _________________________________ _________________________________ Prof. Giovani Lopes Vasconcelos Prof. Clécio Clemente de Souza e Silva Presidente e Orientador _________________________________ Prof. Darren Greg Crowdy Ao meu avˆo, Mauricio do Nascimento. Por ter me mostrado as estrelas, os planetas, os peixes, os matos e os seixos. Por ter me nutrido com curiosidade para uma vida inteira. Por ter me ensinado, no meio de andan¸cas, conversas e muitas piadas, todas as coisas importantes. AGRADECIMENTOS Agrade¸co. Aos meus pais, Marco e Thelma, por me amarem incondicionalmente, e `as minhas av´os, Maria Izabel e Maria do Socorro, pela torcida e est´ımulo constantes. A Deus por estar sempre comigo. Ao meu orientador, Prof. Giovani Vasconcelos, pela confian¸ca que sempre depositou em mim, pela disponibilidade constante em me ajudar, pela liberdade que me concedeu e que fez com que esse trabalho fosse o produto final de um cotidiano amistoso e agrad´avel. Aos meus amigos da faculdade e da vida, em especial a Erton, Walter, Saulo, Rafael e Cear´a(vulgo Tiago), por tudo de excelente que vivemos juntos nesses ´ultimos seis(!) anos. A Moacyr, pela amizade sincera, pelo caf´e, pelas discuss˜oes, pela literatura e pelo sonho (ainda n˜ao realizado) de construir um laborat´orio de dinˆamica de fluidos no terceiro andar! Pela utopia, sobretudo. Ao CNPq pelo apoio financeiro. iv Dear sir or madam, will you read my book? It took me years to write, will you take a look? —THE BEATLES (Paperback Writer) RESUMO Nessa disserta¸c˜ao estudamos a dinˆamica de v´ortices pr´oximos a fronteiras s´olidas em um fluido ideal, atrav´es do modelo de v´ortices puntiformes. Obtivemos as configura¸c˜oes estacion´arias de v´ortices na presen¸ca de um cilindro circular colocado em um escoamento uniforme e investigamos suas propriedades de estabilidade sob pequenas perturba¸c˜oes. Dois sistemas distintos foram estudados. Consideramos inicialmente o caso cl´assico de um cilindro circular colocado em um escoamento uniforme ilimitado. Nesse caso, como se sabe, um par de v´ortices com sentidos opostos ´eobservado na esteira do cilindro, para n´umeros de Reynolds at´ecerca de 50, ao passo que para n´umeros de Reynolds maiores, essa configura¸c˜ao torna-se inst´avel dando lugar `aemiss˜ao alternada de v´ortices. Este sistema foi tratado analiticamente pela primeira vez, atrav´es de um modelo de v´ortices puntiformes, por F¨oppl em 1913. Na primeira parte dessa disserta¸c˜ao, o modelo de F¨oppl ´erevisto e v´arias caracter´ısticas novas desse sistema s˜ao apresentadas, incluindo a existˆencia de um ponto de sela nilpotente no infinito, at´e ent˜ao n˜ao percebido, cujas ´orbitas homocl´ınicas definem a regi˜ao de estabilidade n˜ao-linear do chamado equil´ıbrio de F¨oppl. Al´em disso, estudamos tamb´em a dinˆamica n˜ao-linear resultante de perturba¸c˜oes anti-sim´etricas do equil´ıbrio de F¨oppl e discutimos sua relevˆancia para a emiss˜ao alternada de v´ortices. Na segunda parte, consideramos o movimento de um v´ortice em torno de um cilindro circular colocado acima de uma parede plana infinita. Em experimentos com esse arranjo, um v´ortice estacion´ario ´eobservado na frente do cilindro, uma situa¸c˜ao que n˜ao ´eencontrada no caso cl´assico (i.e., sem o plano). Para estudar a dinˆamica de v´ortices nessa situa¸c˜ao, a regi˜ao do fluido ´einicialmente mapeada em um anel em um plano complexo auxiliar, e o potencial complexo correspondente ´e ent˜ao obtido em termos da chamada fun¸c˜ao prima de Schottky-Klein, que neste caso pode ser escrita em termos de fun¸c˜oes el´ıpticas. As configura¸c˜oes estacion´arias s˜ao ent˜ao calculadas e suas propriedades de estabilidade s˜ao determinadas. Discutimos tamb´em, como as solu¸c˜oes do modelo de v´ortice puntiforme podem ajudar a explicar as observa¸c˜oes experimentais envolvendo a forma¸c˜ao de v´ortices na frente de um cilindro colocado pr´oximo a um plano. Palavras-chave: dinˆamica de v´ortices, v´ortice puntiforme, dinˆamica hamiltoniana, par de F¨oppl, dom´ınio multiplamente conexo, Schottky-Klein. vi ABSTRACT In this thesis the dynamics of vortices near solid boundaries in an ideal fluid is studied using the point vortex model. Stationary configurations of vortices in the presence of a circular cylinder placed in a uniform stream are obtained and their stability properties under small disturbances are investigated. Two different systems are studied. First, the classical case of a circular cylinder placed in a uniform stream in an otherwise unbounded domain is considered. As is well known, in this case a pair of counter-rotating eddies is observed downstream of the cylinder for Reynolds numbers up to about 50, whereas for larger Reynolds number this configuration becomes unstable, leading to vortex shedding. This system was first treated analytically using point vortices by F¨oppl in 1913. In the first part of the thesis, the F¨oppl model is revisited and several novel features of this sys- tem are presented, including the existence of a hitherto unnoticed nilpotent saddle point at infinity whose homoclinic orbits define the region of nonlinear stability of the so-called F¨oppl equilibrium. In addition, the nonlinear dynamics resulting from antisymmetric perturbations of the F¨oppl equilibrium is studied and its relevance to vortex shedding is discussed. In the second part, the motion of a vortex around a cylinder placed above an infinite plane wall is considered. In experiments using this arrangement, a stationary eddy is observed in front of the cylinder, a situation that is not found in the classical case (i.e., without the plane). To study the vortex dynamics in this case, the flow domain is first mapped to an annulus in an auxiliary complex plane and the corresponding complex potential is obtained in terms of the so-called Schottky-Klein prime function, which in this case can be written in terms of elliptic functions. The stationary configurations are then calculated and their stability properties are determined. It is also discussed how the solutions of the point vortex model can help to explain the experimental findings for the vortex formation in front of a cylinder placed near a plane. Keywords: vortex dynamics, point vortex, hamiltonian dynamics, F¨oppl pair, multiply connected domain, Schottky-Klein. vii CONTENTS Chapter 1—Introduction 1 Chapter 2—Potential Flows and Vortex Dynamics 8 2.1 BernoulliEquation .............................. 8 2.2 PotentialFlow................................. 9 2.3 ComplexPotential .............................. 11 2.3.1 UniformFlow............................. 13 2.3.2 SourceandSink............................ 14 2.3.3 Dipole ................................. 15 2.3.4 PointVortex.............................. 16 2.3.5 UniformFlowPastaCircularCylinder . 17 2.4 MethodofImages............................... 18 2.4.1 Milne-ThomsonCircleTheorem . 19 2.5 VortexDynamics ............................... 21 2.5.1 TheVorticityEquation. 21 2.5.2 TheVortexDynamicsTheorems. 22 2.6 ThePointVortexModel ........................... 23 2.6.1 EquationsofMotion ......................... 24 2.6.2 TheEffectivePotential . 25 2.6.2.1 Vortexinauniformflow . 25 2.6.2.2 Vortexnearaplanewall . 26 2.6.2.3 Vortexnearacircularcylinder . 26 2.6.3 HamiltonianDynamics . 27 2.6.3.1 Vortexinauniformflow . 28 2.6.3.2 Vortexnearaplanewall . 28 2.6.3.3 Vortexnearacircularcylinder . 28 Chapter 3—Vortex Dynamics around a Cylinder: the F¨oppl System 29 3.1 StatementoftheProblem .........................
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